Cell fate transition (conversion between cell types) is a fundamental process critical for development and disease progression. Gene regulatory networks controlling cell fate transitions often involve positive feedback loops. Recent data suggest that highly interconnected positive feedback loops (defined as ultra- feedback circuit in this proposal) have additional functions, but the current understanding of these networks is incomplete, partly due to the lack of theories and mathematical methods to analyze such complex circuits. Epithelial-mesenchymal transition (EMT), a process in which rigid epithelial cells convert to motile mesenchymal forms, is an example of cell fate transitions that are regulated by ultra- feedback circuits. EMT occurs in both normal and pathological conditions such as metastasis. Recent discoveries suggest two complex cellular properties that make EMT difficult to understand intuitively: the formation of multiple intermediate EMT states and the partial reversibility of EMT. The functions of the ultra-feedback circuits in regulating the two cellular properties are yet to be defined. The goal of the proposed study is to gain deeper understanding of these properties of EMT by developing new methods, models and theories to characterize the ultra-feedback circuits. We will combine real algebraic geometry, stability analysis and numerical methods to identify stable steady states that arise from ultra-feedback systems, and we will apply the method to analyze the EMT spectrum of cell types. We will quantify partially reversible EMT with a new theoretical framework based on information theory and dynamical systems. Theory driven simulations and experiments will be performed to examine how ultra-feedback circuits control reversibility. We will characterize the roles of ultra-feedback circuits in cell motility and proliferation during EMT using multiscale modeling and live-cell imaging. The proposal brings about new methods to analyze a large, emerging family of dynamical systems containing a wide range of network structures, a new theoretical framework for understanding information transmission and retainment, and a new multiscale modeling framework for systems with complex state transitions and multiple sources of stochasticity. The proposed study addresses fundamental questions about the interplay between two important and emerging properties of EMT (its multistate nature and its restricted reversibility) with mathematical innovations, and it will provide critical insights into gene regulations of cell fate transitions during development and disease progression. The success of the project will lead to new quantitative information of EMT and new concepts for better understanding EMT properties and for analyzing other cell fate transitions involving ultra-feedback circuits.